Mathematical Modeling, Analysis, and Optimal Control of Corruption Dynamics

In this paper, a nonlinear deterministic model for the dynamics of corruption is proposed and analysed qualitatively using the stability theory of differential equations. The basic reproduction number with respect to the corruption-free equilibrium is obtained using next-generation matrix method. The conditions for local and global asymptotic stability of corruption-free and endemic equilibria are established. From the analysis using center manifold theory, the model exhibits forward bifurcation. Then, the model was extended by reformulating it as an optimal control problem, with the use of two time-dependent controls to assess the impact of corruption on human population, namely, campaigning about corruption through media and advertisement and exposing corrupted individuals to jail and giving punishment. By using Pontryagin’s maximum principle, necessary conditions for the optimal control of the transmission of corruption were derived. From the numerical simulation, it was found that the integrated control strategy must be taken to fight against corruption.


Introduction
Corruption is an illegal activity carried out for private gain and benefit, by misuse of authority or power by public (government) or private (company) officeholders [1]. Different types of corruption are documented in the literature and include public corruption [2], private corruption [3], pervasive corruption, and arbitrary corruption [4]. Corruption can originate from either the demand side or the supply side [5]. In general, corruption faces a major threat to the rule of law, democracy and human rights, fairness, and social justice, hinders economic development, and brings market economies at risk for their proper and fair functioning [2,6]. It is a serious problem in all countries in the world mainly for developing countries [6]. Even though most countries have anticorruption policies or strategies that are being made to control corruption, it remains an epidemic in the society.
Mathematical models with optimal control analysis are an important tool in understanding the corruption transmission dynamics and in decision-making processes regarding intervention programs for corruption control. In the current research, there are only a few quantitative studies on corrup-tion.The study by Abdulrahman [7] proposed and analysed a deterministic model for corruption in a population. They computed the basic reproduction number (BRN), corruption-free equilibrium point, and endemic equilibrium point. Numerical simulations were carried out, and they revealed that corruption can only be reduced to a manageable level but not totally removed. In [2], the authors developed and analysed a mathematical model for corruption dynamics. They determined the basic reproduction number and corruption-free and endemic equilibrium points. The study by Lemecha [8] proposed a mathematical model for corruption by considering awareness created by anticorruption and counseling in jail. The existence of unique corruption-free and endemic equilibrium points was investigated, and the basic reproduction number was computed. The study in [9] developed the differential equation-based models representing either growth or decay laws for corruption. In [10], a difference equationbased model was framed to measure the level of corruption. The author in [11] considered a game theoretic approach to deal with corruption. A corruption prevention model was formulated by Khan [12] and showed that the complete prevention of corruption is possible if the ratio between rate of dismissal and rate of corruption is equal to one. In [13], the authors developed an SIR model for the corruption dynamics. Furthermore, they extended the model to include optimal control with a single optimal control strategy. They concluded that the level of corruption in society can be reduced if efforts to control corruption are increased and put in place through media/punishments. Nathan and Jackob developed an epidemiological compartment model for corruption in Kenya, mainly considering those who take advantage of office holders and political office holders [14].
In this paper, an optimal control theory was used to study the effectiveness of all possible combinations of two corruption preventive measures, namely, (i) campaigning about corruption through media and advertisement and (ii) exposing the corrupted individuals to jail and giving punishment. For this, we considered a model for the transmission of corruption and incorporated two time-dependent controls. Firstly, the model has been studied with no control and the stability properties have been investigated. Secondly, we consider the control parameters to be time dependent and examine the impact of different combinations of these measures in controlling corruption. We use Pontryagin's maximum principle to derive necessary conditions for the optimal control of corruption.
The paper is organized as follows. Section 2 is devoted for the model description and the underlying assumptions. In Section 3, we carry out mathematical analysis including bifurcation and sensitivity of the model. In Section 4, the optimal control problem is presented. Section 5 is devoted for numerical simulations and discussions of the model. The conclusion is presented in Section 6.

Model Formulation and Description
The total population NðtÞ is divided into five compartments. Those who are susceptible to corruption are susceptible individuals SðtÞ, those who are exposed to a corrupted person but do not perform it are exposed individuals EðtÞ, those who are performing corruption are corrupted individuals CðtÞ, those who stopped doing corruption are recovered individuals RðtÞ, and those who know the badness of corruption and do not perform it permanently are honest individuals HðtÞ at time t ≥ 0. Assume that there is a positive recruitment Π into the susceptible class by birth or immigration. From this class, κ proportion will join the honest subpopulation which never involve in corruption practices irrespective of the circumstances around them. We consider a positive natural death rate μ for all humans at all time under the study. Susceptible individuals will have a contact rate with the corrupted individuals at rate β with a corruption transmission probability p per contact and moved to the exposed class. From these individuals, α progressed at a rate of δ to the corrupted compartment and the remaining proportion will move to the recovered compartments. The corrupted individuals get knowledge about the effect of corruption in jail and move to the recovered subpopulation at a rate of σ. From these recovered individuals, θ moved at a rate of ε to susceptible compartment and the other proportion joins the honesty compartment. All the descriptions of the parameters are listed in Table 1.
With regard to the above considerations, we have the compartmental flow diagram shown in Figure 1. From the flow chart, the model will be governed by the following system of differential equations: with the initial condition Then, using the variation of constant formula the solution of equation (3) at τ is given by

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Moreover, since all the variables are positive in ½0, τ, hence SðτÞ > 0. It can be shown in a similar way that EðτÞ > 0, CðτÞ > 0, RðτÞ > 0, and HðτÞ > 0, which is a contradiction. Hence, τ = ∞. Therefore, all the solution sets are positive for t ≥ 0.

Invariant Region.
We obtained the invariant region, in which the model solution is bounded. To do this, consider the total human population NðtÞ, where NðtÞ = SðtÞ + EðtÞ + CðtÞ + RðtÞ + HðtÞ.
Substituting all state equations from the model (1), we get Integrating equation (6) both sides and taking t ⟶ ∞, we get is positively invariant, inside which the model is considered to be epidemiologically meaningful and mathematically well posed.

Corruption-Free Equilibrium Point (CFEP).
The corruption-free equilibrium point of model (1) is obtained by equating all equations of the model to zero and letting E = 0, C = 0, and R = 0. Then, we get 3.4. Basic Reproduction Number. The basic reproduction number R 0 measures the expected number of secondary infections that result from one newly infected individual introduced into a susceptible population. We calculate R 0 of the model using the next-generation matrix method as described in [15]. The first step to get R 0 is rewriting the model equations starting with newly infective classes: Then, by the principle of next-generation matrix, the Jacobian, at CFEP, we obtained Therefore, the basic reproduction number is the spectral radius of the next generation matrix FV −1 and is given by 3.5. Local Stability of CFEP  Proof. To prove this theorem, let us first find the Jacobian matrix of system (1): Evaluating equation (12) at the corruption-free equilibriumE 0 = ðΠ/ðk + μÞ, 0, 0, 0, Π/μðk + μÞÞ, we got From the Jacobian matrix, a characteristic polynomial was obtained as where ψ 1 = σ + 2μ + δ, From equation (14), we have From the last expression of equation (14), we have The Routh-Hurwitz criteria was applied, and equation (17) has strictly negative real root iff ψ 1 > 0 and ψ 2 > 0. Clearly, we see that ψ 1 > 0 because it is the sum of positive parameters and also Hence, the CFEP is locally asymptotically stable if R 0 < 1. This result implies that corruption can be eliminated if the initial size of the corrupted individuals is in the basin of attraction of the CFEP. On the other hand, to ensure that corruption elimination is independent of the initial population size of the corrupted population, it is necessary to show that the CFEP is globally stable.

Global Stability of CFEP
Proof. Consider the following Lyapunov function Differentiating equation (19) with respect to t gives Substituting dE/dt and dC/dt from the model (1), we get Here, take χ 1 = ðαδ/ðδ + μÞÞχ 2 , then we have Taking χ 2 = 1 and substituting R 0 , we get for C ≤ C 0 = π/ðk + μÞ and dV/dt ≤ 0 for R 0 < 1 and dV/ dt = 0 if and only if C = 0. This implies that the only trajectory of the system (1) on which dV/dt ≤ 0 is E 0 . Therefore, by LaSalle's invariance principle, E 0 is globally asymptotically stable in Ω. This supports the forward bifurcation of Section 3.9.
Then, we obtain where

Local Stability of Endemic Equilibrium
Theorem 4. The endemic equilibrium E * of system (1) is locally asymptotically stable in Ω if R 0 > 1, otherwise unstable.
Proof. Let us first obtain the Jacobian matrix of system (1): The characteristic polynomial of equation (27) is given by where Using the Routh-Hurwitz criterion, all roots of characteristic polynomial have negative real parts if and only if ψ 1 > 0, Hence, the endemic equilibrium E * is locally asymptotically stable.
3.9. Bifurcation Analysis. In order to determine the direction of the bifurcation and to prove the stability of the endemic equilibrium point, we make use of the bifurcation theory, which is based on the center manifold theory [16] (as described by Theorem 6 of Castillo-Chavez and Song [16]).
Theorem 5. The model in the system (1) exhibits forward bifurcation at R 0 = 1 and a unique endemic equilibrium exists.
Proof. To conduct this analysis, we assume that S = z 1 , E = z 2 , C = z 3 , R = z 4 , H = z 5 and from R 0 = 1, we have The new the model in equation (1) is given by
The left eigenvector, v = ðv 1 , v 2 , v 3 , v 4 , v 5 Þ, associated with this simple zero eigenvalue can be obtained from vJ = 0 and given by Since the first, fourth, and fifth components of v are zero, we do not need the derivatives of f 1 , f 4 , and f 5 . From the derivatives of f 2 and f 3 , the only ones that are nonzero are the following: and all the other partial derivatives are zero. The direction of the bifurcation at R 0 = 1 is determined by the signs of the bifurcation coefficients a and b, obtained from the above partial derivatives, given, respectively, by Since the coefficient b is positive and the sign of the coefficient a is negative, the model undergoes forward bifurcation, which implies that R 0 < 1 is a necessary and sufficient condition for corruption elimination.
3.10. Sensitivity Analysis. In order to see the relative effect of each parameter to the corruption transmission, we performed sensitivity analysis. To go through, we followed the approach defined by [17] as done in [18,19] which used the definition of normalized forward sensitivity index defined as a variable, R 0 , that depends differentiably on a parameter,ℓ, defined as for ℓ represents all the basic parameters and R 0 = αpβδΠ /ðδ + μÞðσ + μÞðκ + μÞ.
And we do this in a similar fashion for the remaining parameters.
3.11. Interpretation of Sensitivity Indices. The sensitivity indices of parameters are found in Table 2. Parameters that have positive indices ðβ, α, and δÞ have great impact on expanding the corruption in the community when their Journal of Applied Mathematics values increase. Parameters with negative indices ðκ, σ, and μÞ minimize the burden of corruption in the community as their values increase. Therefore, the model sensitivity analysis demonstrated that policy makers are supposed to decrease positive index parameters and increase negative index parameters to combat corruption in a population. From this result, to identify the optimal strategy to fight against corruption, an optimal control model analysis is considered in the next section.

Extension of the Model into Optimal Control
The optimal level that would be needed to control corruption is minimizing the number of exposed and corrupted population. An optimal control model was developed by including the following two controls: (i) u 1 : campaigning about corruption through media and advertisement (ii) u 2 : exposing the corrupted individuals to jail and giving punishment Adding the two controls on the model (1), the optimal control model is given by The control variables u 1 and u 2 minimize the optimal control model (40) subject to the objective functional defined as where t f is the final time, a 1 and a 2 are weight constants of the exposed and corrupted population, respectively, while w 1 and w 2 are weight coefficients for each individual control measure. We choose a nonlinear cost on the controls based on the assumption that the cost takes nonlinear form [20]. Optimal control function ðu * 1 , u * 2 Þ needs to be found such that where U = fðu 1 , u 2 Þju i ðtÞ is lebesgue measurable on ½0, t f , 0 ≤ u i ðtÞ ≤ 1, i = 1, 2g is the closed set.

The Hamiltonian and Optimality
System. The optimal control must satisfy the necessary conditions that are formulated by Pontryagin's maximum principle ( [21]). This principle converts the system of equations (40) and (41) into a problem of minimizing point-wise a Hamiltonian (M), with respect to u 1 ðtÞ and u 2 ðtÞ as where λ i , i = 1, ⋯, 5 are the adjoint variable functions to be determined. Because of the convexity of the integrand of J with respect to u 1 and u 2 , a priori boundaries of the state solutions and the Lipschitz property of the state system relating to the state variables, the existence of the optimal control was proved using Corollary 4.1 in the work of Fleming and Rishel [22].

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Proof. The adjoint equation and transversality conditions are standard results obtained from Pontryagin's maximum principle [21]. Differentiate the Hamiltonian with respect to states S, E, C, R, and H, respectively, then we got the adjoint equations as With transversality conditions, λiðtf Þ = 0, i = 1, ⋯, 5. Differentiate the Hamiltonian equation with respect to u 1 and u 2 and obtain the characterization of optimal controls u * 1 , u * 2 from ∂M/∂u 1 = 0, ∂M/∂u 2 = 0. Thus, we got the characteristic equation in standard control arguments form involving the bounds on the controls as follows: In compacted notation: where The optimality system is formed from the optimal control system (the state system) and the adjoint variable system by incorporating the characterized control set under initial and transversal condition: u * 1 = max 0, min 1, ,

Numerical Simulations and Discussions
In this section, numerical simulations are performed on the model (1). Maple 18 software is used to check the effect of some parameters in the spread and control of corruption. The  Table 3   Journal of Applied Mathematics keeping the other parameters constant. It is evident from Figure 2(b) that, when the value of δ increases, the number of corrupted population also increases. Therefore, public policy makers must concentrate on minimizing the contact rate of corrupted individuals to susceptible population and reduce the transition of exposed individuals to the corrupted population.

5.2.
Effect of κ and σ on the Corrupted Population. In Figure 3(a), we can see that κ plays an important role in reducing the number of corrupted population. When the rate at which individuals join the honest subpopulation from susceptible class increases from 0.03 to 0.4, the number of corrupted population decreases. From Figure 3(b), the rate at which corrupted individuals join the recovered population through education and/or jail sentence increases from 0.007 to 0.1; the number of corrupted population decreases. Therefore, public policy makers must concentrate on maximizing the number of populations that join the honest subpopulation and corrupted population to recover from doing it.
To simulate the optimality system in (50), iterative technique is applied which gives the optimal solution. Forward fourth-order Runge-Kutta method is used to solve the state system, and backward fourth-order Runge-Kutta method is used to solve the adjoint system by considering the initial condition of the state system and the final condition of the  11 Journal of Applied Mathematics adjoint systems. The controls continue to be updated by combining from the previous result of the controls with the characterization. The solution of the state and adjoint system is repeated by the updated controls. This condition continues repeatedly until consecutive iteration is close enough each other [23]. We proposed the following three strategies for numerical simulation of the corruption model: In this strategy, controls u 1 were used to optimize the objective function J while we set control u 2 to zero. In Figure 4, we see that exposed and corrupted population significantly reduced when there is control compared to situation with no control. From this, we conclude that applying these strategy can reduce corruption along with time from the community but not possible to avoid it in a specified period of time. Hence, it is not an effective strategy to combat corruption from the population.

Strategy B:
Exposing the Corrupted Individuals to Jail and Giving Punishment. Under this strategy, we optimize the objective function J using the control u 2 while u 1 is set to zero. It is evident from Figure 5 that the number of exposed individuals cannot reduce using the strategy and the corrupted population seems eliminated in the first year but grows in number due to the increase in number of exposed individuals. Therefore, the strategy is not effective in fighting and reducing corruption in the population even though it reduces the number of corrupted individuals when compared with control from without control.

Strategy C: Combination of Both Strategies A and B.
Under this strategy, both the control measures ðu 1 and u 2 Þ are used to optimize the objective functional J. The results of the applications of the strategy are shown in Figure 6. It can be seen from Figure 6 that the exposed and corrupted populations decrease as compared to no control. Therefore, the intervention strategy is effective in bringing down the exposed and corrupted population in the specified period of time.

Conclusions
In this paper, a mathematical model for the transmission dynamics of corruption in a population was formulated. The basic reproduction number R 0 was computed, and the stability of equilibrium points was investigated. Through Lyapunov's theory, the corruption-free equilibrium point is globally asymptotically stable whenever R 0 < 1 was proven. When R 0 = 1, bifurcation analysis of the model was proven and it exhibits forward bifurcation. The local stability of the unique endemic equilibrium whenever R 0 > 1 was demonstrated. Using the definition of normalized forward sensitivity, the sensitivity parameters were determined. It can be inferred that positive indices should be decreased while increasing negative indices to minimize/control corruption in a population. Then, the optimal control model was formulated by adding two time-dependent controls (campaigning about corruption through media and advertisement and exposing the corrupted individuals to jail and giving

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Journal of Applied Mathematics punishment). Optimal control theory was used to establish conditions under which the spread of corruption can be stopped and to examine the impact of a possible combination of these two controls on the transmission of corruption. The characterization of the optimal control was obtained by the application of the Pontryagin's maximum principle. It is evident from the results of the numerical simulations that the integrated control strategy should be taken to fight against corruption. The cost-effectiveness analysis of the integrated strategy would be considered in a future study.

Data Availability
The data supporting this deterministic model are from previous published articles and they have been duly cited in this paper.

Conflicts of Interest
The author declares that there is no conflict of interest regarding the publication of this paper.